When your toilet only flushes part of the way, the problem lies in the toilet itself and not your plumbing. The stopper inside the toilet must stay open long enough to allow water from the top of the toilet to flow through the hole and into the bottom of the toilet. If this hole does not open for enough time, the toilet will not receive enough water to complete a full flush.
Take the lid off the top of the toilet and carefully set it aside.
Remove the lift chain from the stopper at the bottom of the basin. The stopper is a rubber circle about half the size of a CD.
Insert a link on a lower part of the chain into the connection hook at the top of the stopper.
Look at how the chain hangs. It should hang straight down with about 1/2 inch slack. If it is already hanging with 1/2 inch slack, lift up the stopper to make sure no debris is caught between it and the bottom of the basin. Even a small air gap between the bottom of the basin and the stopper can cause water to flow out of the toilet consistently. When water consistently flows out, the basin may not retain enough water to fully flush.
Flush the toilet once to check it. The toilet should fully flush. If it does not, wait a few moments and allow the basin to fill and flush it again. Make sure there is not so much chain slack that the slack falls between the stopper and the basin when you flush.